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Mirrors > Home > MPE Home > Th. List > Mathboxes > lsatlspsn | Structured version Visualization version GIF version |
Description: The span of a nonzero singleton is an atom. (Contributed by NM, 16-Jan-2015.) |
Ref | Expression |
---|---|
lsatset.v | ⊢ 𝑉 = (Base‘𝑊) |
lsatset.n | ⊢ 𝑁 = (LSpan‘𝑊) |
lsatset.z | ⊢ 0 = (0g‘𝑊) |
lsatset.a | ⊢ 𝐴 = (LSAtoms‘𝑊) |
lsatlspsn.w | ⊢ (𝜑 → 𝑊 ∈ LMod) |
lsatlspsn.x | ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) |
Ref | Expression |
---|---|
lsatlspsn | ⊢ (𝜑 → (𝑁‘{𝑋}) ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lsatlspsn.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) | |
2 | eqid 2610 | . . 3 ⊢ (𝑁‘{𝑋}) = (𝑁‘{𝑋}) | |
3 | sneq 4135 | . . . . . 6 ⊢ (𝑣 = 𝑋 → {𝑣} = {𝑋}) | |
4 | 3 | fveq2d 6107 | . . . . 5 ⊢ (𝑣 = 𝑋 → (𝑁‘{𝑣}) = (𝑁‘{𝑋})) |
5 | 4 | eqeq2d 2620 | . . . 4 ⊢ (𝑣 = 𝑋 → ((𝑁‘{𝑋}) = (𝑁‘{𝑣}) ↔ (𝑁‘{𝑋}) = (𝑁‘{𝑋}))) |
6 | 5 | rspcev 3282 | . . 3 ⊢ ((𝑋 ∈ (𝑉 ∖ { 0 }) ∧ (𝑁‘{𝑋}) = (𝑁‘{𝑋})) → ∃𝑣 ∈ (𝑉 ∖ { 0 })(𝑁‘{𝑋}) = (𝑁‘{𝑣})) |
7 | 1, 2, 6 | sylancl 693 | . 2 ⊢ (𝜑 → ∃𝑣 ∈ (𝑉 ∖ { 0 })(𝑁‘{𝑋}) = (𝑁‘{𝑣})) |
8 | lsatlspsn.w | . . 3 ⊢ (𝜑 → 𝑊 ∈ LMod) | |
9 | lsatset.v | . . . 4 ⊢ 𝑉 = (Base‘𝑊) | |
10 | lsatset.n | . . . 4 ⊢ 𝑁 = (LSpan‘𝑊) | |
11 | lsatset.z | . . . 4 ⊢ 0 = (0g‘𝑊) | |
12 | lsatset.a | . . . 4 ⊢ 𝐴 = (LSAtoms‘𝑊) | |
13 | 9, 10, 11, 12 | islsat 33296 | . . 3 ⊢ (𝑊 ∈ LMod → ((𝑁‘{𝑋}) ∈ 𝐴 ↔ ∃𝑣 ∈ (𝑉 ∖ { 0 })(𝑁‘{𝑋}) = (𝑁‘{𝑣}))) |
14 | 8, 13 | syl 17 | . 2 ⊢ (𝜑 → ((𝑁‘{𝑋}) ∈ 𝐴 ↔ ∃𝑣 ∈ (𝑉 ∖ { 0 })(𝑁‘{𝑋}) = (𝑁‘{𝑣}))) |
15 | 7, 14 | mpbird 246 | 1 ⊢ (𝜑 → (𝑁‘{𝑋}) ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 = wceq 1475 ∈ wcel 1977 ∃wrex 2897 ∖ cdif 3537 {csn 4125 ‘cfv 5804 Basecbs 15695 0gc0g 15923 LModclmod 18686 LSpanclspn 18792 LSAtomsclsa 33279 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-fv 5812 df-lsatoms 33281 |
This theorem is referenced by: lsatspn0 33305 dvh4dimlem 35750 dochsnshp 35760 lclkrlem2a 35814 lclkrlem2c 35816 lclkrlem2e 35818 lcfrlem20 35869 mapdrvallem2 35952 mapdpglem20 35998 mapdpglem30a 36002 mapdpglem30b 36003 hdmaprnlem3eN 36168 hdmaprnlem16N 36172 |
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